RTSJ,
Of the three you mention, my own choice would be the 7mm, IF no other options were to be considered.
I have taken, and used, several rifles on 4 safaris to Africa, including my .300 Weatherby on the first trip to Namibia, my guide's 7x57 for a number of plains game as well in S. Africa, a .375 Ruger for everything from Springbok to Lion, and on my last hunt, a .30-06, with 165 gr. Barnes TSX bullets, for the tiny antelopes.
Were I to return to Africa for another plains game hunt, for anything and everything from duiker to eland, my choice would unquestionably be the .30-06, again. In my own personal experience, nothing is as versatile or capable for any and every situation you might encounter- and with a bullet like the TSX, you are ready for any game that might be presented. Ammo for the '06 is likely to be more available than for most calibers you might be using, as well. My choice of rifle was a Browning X-Bolt, topped with a Leupold 2.5-8x VX3 in Talley LW mounts, all-up weight of 7.5 lbs- a real joy to carry around, and crazy accurate with the Barnes bullet.
You don't need the RUM for any of the species you mention, and in general, long range shooting is not a necessity.
If you have lots of time before your trip, enjoy the preparation- it's half the fun of the experience! I would also recommend Craig Boddington's book, 'Safari Rifles II', as an excellent read regarding all things safari rifle related.
